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Basically Tian's invariant relies fundamentally on a lemma of Homander which says that $e^{- \phi}$ is integrable for $\phi$ plurisubharmonic on a ball of radius $1$ under some extra assumption. I am reading through a detailed proof of it on this note, p15, proof of theorem 3.10. I am confused as to why it is necessary to choose geodesic balls to cover the manifold, and it is not clear to me where the property of geodesic balls are used in the proof. Any clarification/help on this is much appreciated!

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    $\begingroup$ it's not important, you can cover $X$ by any (small enough) charts. $\endgroup$
    – Henri
    Sep 2, 2020 at 19:04
  • $\begingroup$ Thank you. I just want to make sure that I am not missing any important information here. $\endgroup$
    – Linda Lee
    Sep 2, 2020 at 19:44

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